Determine Dumbbell Distribution: Solving for 7.1 kg Weights Using Averages

To solve for the number of dumbbells weighing 7.1 kg, we will leverage the weighted average given by the problem:

• Calculate weights of known dumbbells:
  • Total weight of 3 dumbbells at 4.5 kg each: $3 \times 4.5 = 13.5$ kg
  • Total weight of 4 dumbbells at 5.2 kg each: $4 \times 5.2 = 20.8$ kg
• Let $x$ be the number of dumbbells weighing 7.1 kg. Therefore, the remaining $10 - x$ dumbbells weigh 3.8 kg each.
• Calculate total weight: $13.5 + 20.8 + 7.1x + 3.8(10 - x) = 5.22 \times 17$ Simplifying the right, we have: $13.5 + 20.8 + 7.1x + 38 - 3.8x = 88.74$ $7.1x - 3.8x + 13.5 + 20.8 + 38 = 88.74$ $3.3x + 72.3 = 88.74$ $3.3x = 88.74 - 72.3$ $3.3x = 16.44$ $x = \frac{16.44}{3.3} = 5$

Therefore, the number of dumbbells weighing 7.1 kg is $5$.